IAPWS R7-97(2012)
The International Association for the Properties of Water and Steam
Lucerne, Switzerland
August 2007
Revised Release on the IAPWS Industrial Formulation 1997
for the Thermodynamic Properties of Water and Steam
(The revision only relates to the extension of region 5 to 50 MPa)
2007 International Association for the Properties of Water and Steam
Publication in whole or in part is allowed in all countries provided that attribution is given to the
International Association for the Properties of Water and Steam
President:
J. R. Cooper
School of Engineering and Materials Science
Queen Mary, University of London
Mile End Road
London E1 4NS, England
Executive Secretary:
Dr. R. B. Dooley
Structural Integrity Associates, Inc.
2616 Chelsea Drive
Charlotte, NC 28209, USA
email: bdooley@structint.com
This revised release replaces the corresponding release of 1997 and contains 49 pages, including this
cover page.
This release has been authorized by the International Association for the Properties of Water and
Steam (IAPWS) at its meeting in Lucerne, Switzerland, 26-31 August, 2007, for issue by its
Secretariat. The members of IAPWS are: Argentina and Brazil, Britain and Ireland, Canada, the Czech
Republic, Denmark, France, Germany, Greece, Italy, Japan, Russia, and the United States of America,
and associate member Switzerland.
The formulation provided in this release is recommended for industrial use, and is called "IAPWS
Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam" abbreviated to
"IAPWS Industrial Formulation 1997" (IAPWS-IF97). The IAPWS-IF97 replaces the previous
Industrial formulation "The 1967 IFC Formulation for Industrial Use" (IFC-67) [1]. Further details
about the formulation can be found in the article by W. Wagner et al. [2] and for the extended region 5
in [2a]. Additional supplementary backward equations have been adopted by IAPWS and are listed in
[2b].
The material contained in this release is identical to that contained in the release on IAPWS-IF97,
issued by IAPWS in September 1997, except for the basic equation for region 5. The previous basic
equation for this region was replaced by a new equation of the same structure. For temperatures
between 1073 K and 2273 K, this equation extends the upper range of validity of IAPWS-IF97 in
pressure from 10 MPa to 50 MPa. Except for the basic equation for region 5 of IAPWS-IF97, the
property calculations are unchanged.
In addition, minor editorial changes to this document, involving correction of typographical errors
and updating of references, were made in the 2007 revision, and again in 2009 and 2010. In 2012, the
description of the range of validity of the equation for the metastable vapor was clarified.
IAPWS also has a formulation intended for general and scientific use [3].
Further information about this release and other releases issued by IAPWS can be obtained from
the Executive Secretary of IAPWS or from http://www.iapws.org.
2
Contents
1 Nomenclature
3
2 Structure of the Formulation
4
3 Reference Constants
5
4 Auxiliary Equation for the Boundary between Regions 2 and 3
5
5 Equations for Region 1
6
5.1 Basic Equation
5.2 Backward Equations
5.2.1 The Backward Equation T ( p,h )
5.2.2 The Backward Equation T ( p,s )
6 Equations for Region 2
6
9
10
11
12
6.1 Basic Equation
13
6.2 Supplementary Equation for the Metastable-Vapor Region
6.3 Backward Equations
6.3.1 The Backward Equations T( p,h ) for Subregions 2a, 2b, and 2c
6.3.2 The Backward Equations T( p,s ) for Subregions 2a, 2b, and 2c
17
20
22
25
7 Basic Equation for Region 3
29
8 Equations for Region 4
33
8.1 The Saturation-Pressure Equation (Basic Equation)
8.2 The Saturation-Temperature Equation (Backward Equation)
33
35
9 Basic Equation for Region 5
36
10 Consistency at Region Boundaries
40
10.1 Consistency at Boundaries between Single-Phase Regions
10.2 Consistency at the Saturation Line
11 Computing Time of IAPWS-IF97 in Relation to IFC-67
11.1 Computing-Time Investigations for Regions 1, 2, and 4
11.2 Computing-Time Investigations for Region 3
40
41
43
43
45
12 Estimates of Uncertainties
46
13 References
49
3
1 Nomenclature
Thermodynamic quantities:
Superscripts:
cp
cv
Specific isobaric heat capacity
Specific isochoric heat capacity
o
r
Ideal-gas part
Residual part
f
g
h
M
p
R
Rm
s
T
u
v
w
x
Specific Helmholtz free energy
Specific Gibbs free energy
Specific enthalpy
Molar mass
Pressure
Specific gas constant
Molar gas constant
Specific entropy
Absolute temperature a
Specific internal energy
Specific volume
Speed of sound
General quantity
Transformed pressure, Eq. (29a)
Dimensionless Gibbs free energy,  = g /(RT )
Reduced density,  =  /*
Difference in any quantity
Reduced enthalpy,  = h / h*
Reduced temperature,  = T / T *
Transformed temperature, Eq. (29b)
Reduced pressure, = p / p*
Mass density
Reduced entropy,  = s / s*
Inverse reduced temperature,  = T */ T
Dimensionless Helmholtz free energy,  = f /(RT )
*
Reducing quantity
Saturated liquid state
Saturated vapor state












a


Subscripts:
c
max
RMS
s
t
tol
Critical point
Maximum value
Root-mean-square value
Saturation state
Triple point
Tolerance of a quantity
Root-mean-square deviation:
xRMS 
1
N
N
 ( x n ) 2
n 1
where xn can be either absolute
or percentage difference between
the corresponding quantities x ; N
is the number of xn values
(depending on the property,
between 10 million and 100
million points are uniformly
distributed over the respective
range of validity).
Note: T denotes absolute temperature on the International Temperature Scale of 1990.
4
2 Structure of the Formulation
The IAPWS Industrial Formulation 1997 consists of a set of equations for different regions
which cover the following range of validity:
273.15 K  T  1073.15 K
1073.15 K  T  2273.15 K
p  100 MPa
p  50 MPa .
Figure 1 shows the five regions into which the entire range of validity of IAPWS-IF97 is
divided. The boundaries of the regions can be directly taken from Fig. 1 except for the
boundary between regions 2 and 3; this boundary is defined by the so-called B23-equation
given in Section 4. Both regions 1 and 2 are individually covered by a fundamental equation
for the specific Gibbs free energy g( p,T ), region 3 by a fundamental equation for the specific
Helmholtz free energy f ( ,T ), where  is the density, and the saturation curve by a
saturation-pressure equation ps (T). The high-temperature region 5 is also covered by a g( p,T )
equation. These five equations, shown in rectangular boxes in Fig. 1, form the so-called basic
equations.
Fig. 1. Regions and equations of IAPWS-IF97.
Regarding the main properties specific volume v, specific enthalpy h, specific isobaric heat
capacity cp , speed of sound w, and saturation pressure ps , the basic equations represent the
corresponding values from the "IAPWS Formulation 1995 for the Thermodynamic Properties
of Ordinary Water Substance for General and Scientific Use" [3] (hereafter abbreviated to
IAPWS-95) to within the tolerances specified for the development of the corresponding
equations; details of these requirements and their fulfillment are given in the comprehensive
paper on IAPWS-IF97 [2]. The basic equations for regions 1 and 3 also yield reasonable
values for the metastable states close to the stable regions. For region 2 there is a special
5
equation for the metastable-vapor region. Along the region boundaries the corresponding
basic equations are consistent with each other within specified tolerances; for details see
Section 10.
In addition to the basic equations, for regions 1, 2, and 4 so-called backward equations are
provided in the forms of T ( p,h ) and T ( p,s ) for regions 1 and 2, and Ts( p ) for region 4.
These backward equations are numerically consistent with the corresponding basic equations
and make the calculation of properties as functions of p,h and of p,s for regions 1 and 2, and
of p for region 4, extremely fast. In this way, properties such as T ( p,h ), h ( p,s ), and h( p ) can
be calculated without any iteration from the backward equation alone or by combination with
the corresponding basic equation, for example, h ( p,s ) via the relation h ( p,T ( p,s )). As a
consequence, the calculation of the industrially most important properties is on average more
than five times faster than the corresponding calculation with IFC-67; for details see
Section 11.
The estimates of uncertainty of the most relevant properties calculated from the
corresponding equations of IAPWS-IF97 are summarized in Section 12.
3 Reference Constants
The specific gas constant of ordinary water used for this formulation is
R = 0.461 526 kJ kg K.
(1)
This value results from the recommended values of the molar gas constant [4], and the molar
mass of ordinary water [5, 6]. The values of the critical parameters
Tc = 647.096 K
pc = 22.064 MPa
c = 322 kg m
(2)
(3)
(4)
are from the corresponding IAPWS release [7].
4 Auxiliary Equation for the Boundary between Regions 2 and 3
The boundary between regions 2 and 3 (see Fig. 1) is defined by the following simple
quadratic pressure-temperature relation, the B23-equation
  n1  n2  n3 2
(5)
where  = p/p* and  = T / T * with p* = 1 MPa and T * = 1 K. The coefficients n1 to n3 of
Eq. (5) are listed in Table 1. Equation (5) roughly describes an isentropic line; the entropy
values along this boundary line are between s = 5.047 kJ kg K and s = 5.261 kJ kg K.
6
Alternatively Eq. (5) can be expressed explicitly in temperature as
1/2
  n4    n5  / n3 
(6)
,
with  and  defined for Eq. (5) and the coefficients n3 to n5 listed in Table 1. Equations (5)
and (6) cover the range from 623.15 K at a pressure of 16.5292 MPa to 863.15 K at 100 MPa.
Table 1. Numerical values of the coefficients of the B23-equation, Eqs. (5) and (6), for
defining the boundary between regions 2 and 3
i
ni
i
1
0.348 051 856 289 69 × 103
4
0.572 544 598 627 46 × 103
2
–0.116 718 598 799 75 × 101
5
0.139 188 397 788 70 × 102
3
ni
0.101 929 700 393 26 × 10
For computer-program verification, Eqs. (5) and (6) must meet the following T-p point:
T = 0.623 150 000  103 K , p = 0.165 291 643  102 MPa.
5 Equations for Region 1
This section contains all details relevant for the use of the basic and backward equations of
region 1 of IAPWS-IF97. Information about the consistency of the basic equation of this
region with the basic equations of regions 3 and 4 along the corresponding region boundaries
is summarized in Section 10. The results of computing-time comparisons between
IAPWS-IF97 and IFC-67 are given in Section 11. The estimates of uncertainty of the most
relevant properties can be found in Section 12.
5.1 Basic Equation
The basic equation for this region is a fundamental equation for the specific Gibbs free
energy g. This equation is expressed in dimensionless form,  = g/(RT ), and reads
34
g( p, T )
I
J
   ,    ni  7.1    i   1.222  i ,
RT
i 1
(7)
where  = p/p* and  = T */T with p* = 16.53 MPa and T * = 1386 K ; R is given by Eq. (1).
The coefficients ni and exponents Ii and Ji of Eq. (7) are listed in Table 2.
All thermodynamic properties can be derived from Eq. (7) by using the appropriate
combinations of the dimensionless Gibbs free energy and its derivatives. Relations between
the relevant thermodynamic properties and  and its derivatives are summarized in Table 3.
All required derivatives of the dimensionless Gibbs free energy are explicitly given in
Table 4.
7
Since the 5th International Conference on the Properties of Steam in London in 1956, the
specific internal energy and the specific entropy of the saturated liquid at the triple point have
been set equal to zero:
ut  0 ; st  0 .
(8)
In order to meet this condition at the temperature and pressure of the triple point
Tt = 273.16 K [8]
pt = 611.657 Pa [9] ,
(9)
the coefficients n3 and n4 in Eq. (7) have been adjusted accordingly. As a consequence, Eq. (7)
yields for the specific enthalpy of the saturated liquid at the triple point
ht = 0.611 783 J kg1 .
(10)
Table 2. Numerical values of the coefficients and exponents of the dimensionless Gibbs free energy
for region 1, Eq. (7)
i
Ii
Ji
i
Ii
Ji
1
0
–2
0.146 329 712 131 67
18
2
3
2
0
–1
– 0.845 481 871 691 14
19
2
17
– 0.726 949 962 975 94 × 10–15
3
0
0
– 0.375 636 036 720 40 × 101
20
3
–4
– 0.316 796 448 450 54 × 10–4
4
0
1
0.338 551 691 683 85 × 101
21
3
0
– 0.282 707 979 853 12 × 10–5
5
0
2
– 0.957 919 633 878 72
22
3
6
– 0.852 051 281 201 03 × 10–9
6
0
3
0.157 720 385 132 28
23
4
–5
– 0.224 252 819 080 00 × 10–5
7
0
4
– 0.166 164 171 995 01 × 10–1
24
4
–2
– 0.651 712 228 956 01 × 10–6
8
0
5
0.812 146 299 835 68 × 10–3
25
4
10
– 0.143 417 299 379 24 × 10–12
9
1
–9
0.283 190 801 238 04 × 10–3
26
5
–8
– 0.405 169 968 601 17 × 10–6
10
1
–7
– 0.607 063 015 658 74 × 10–3
27
8
–11
– 0.127 343 017 416 41 × 10–8
11
1
–1
– 0.189 900 682 184 19 × 10–1
28
8
–6
– 0.174 248 712 306 34 × 10–9
12
1
0
– 0.325 297 487 705 05 × 10–1
29
21
–29
– 0.687 621 312 955 31 × 10–18
13
1
1
– 0.218 417 171 754 14 × 10–1
30
23
–31
0.144 783 078 285 21 × 10–19
14
1
3
– 0.528 383 579 699 30 × 10–4
31
29
–38
0.263 357 816 627 95 × 10–22
15
2
–3
– 0.471 843 210 732 67 × 10–3
32
30
–39
– 0.119 476 226 400 71 × 10–22
16
2
0
– 0.300 017 807 930 26 × 10–3
33
31
–40
0.182 280 945 814 04 × 10–23
17
2
1
0.476 613 939 069 87 × 10–4
34
32
–41
– 0.935 370 872 924 58 × 10–25
ni
ni
– 0.441 418 453 308 46 × 10–5
8
Table 3. Relations of thermodynamic properties to the dimensionless Gibbs
free energy and its derivatives a when using Eq. (7)
Property
Relation
Specific volume
v( , )
Specific internal energy
u  ,  
v   g/ p T
u  g  T  g/T  p  p(g/ p)T
p
  
RT
RT
Specific entropy
s  ,  
s    g/T  p
R
h  ,  
Specific enthalpy
h  g  T  g/T  p
RT
Specific isobaric heat capacity
cp
c p   h /T  p
cv
cv   u /T v
    
  
 ,  
R
Specific isochoric heat capacity
      
  2  
 ,    2
R
Speed of sound


 2
w2 ( , )

RT
(      )2
w  v   ( p / v) s 
1/2
 2 
a
(      )2
 
  
  2 
  2 
  2 
 
 
     ,     2  ,      ,     2  ,    

  
  
   
   
   
Table 4. The dimensionless Gibbs free energy  and its derivatives a according to Eq. (7)
34
   ni  7.1    i   1.222 
I
Ji
i 1
34
I 1
     ni I i  7.1    i
i 1
 
34
  1.222 Ji
 ni  7.1    i Ji   1.222 
I
i 1
34
Ji 1
34
i 1
  
I 1
     ni I i  7.1    i
i 1
a
I 2
    ni I i  I i  1  7.1    i
34
 ni  7.1   
Ii
i 1
J i   1.222 
  1.222 Ji
J i  J i  1   1.222 
Ji 1
  2 
  2 
  2 
  
  
     ,     2  ,      ,     2  ,    

  
  
  
  
   
J i 2
9
Range of validity
Equation (7) covers region 1 of IAPWS-IF97 defined by the following range of
temperature and pressure; see Fig. 1:
273.15 K  T  623.15 K
ps(T)  p  100 MPa .
In addition to the properties in the stable single-phase liquid region, Eq. (7) also yields
reasonable values in the metastable superheated-liquid region close to the saturated liquid line.
Note: For temperatures between 273.15 K and 273.16 K, the part of the range of validity between the pressures
on the melting line [10] and on the saturation-pressure line, Eq. (30), corresponds to metastable states.
Computer-program verification
To assist the user in computer-program verification of Eq. (7), Table 5 contains test values
of the most relevant properties.
Table 5. Thermodynamic property values calculated from Eq. (7) for selected values of T and p a
T = 300 K,
p = 3 MPa
T = 300 K,
p = 80 MPa
T = 500 K,
p = 3 MPa
v / (m3 kg–1)
0.100 215 168 × 10–2
0.971 180 894 × 10–3
0.120 241 800 × 10–2
h / (kJ kg–1)
0.115 331 273 × 103
0.184 142 828 × 103
0.975 542 239 × 103
u / (kJ kg–1)
0.112 324 818 × 103
0.106 448 356 × 103
0.971 934 985 × 103
s / (kJ kg–1 K–1)
0.392 294 792
0.368 563 852
0.258 041 912 × 101
cp / (kJ kg–1 K–1)
0.417 301 218 × 101
0.401 008 987 × 101
0.465 580 682 × 101
w / (m s–1)
0.150 773 921 × 104
0.163 469 054 × 104
0.124 071 337 × 104
a It is recommended to verify programmed functions using 8 byte real values for all three combinations of T and
p given in this table.
5.2 Backward Equations
For the calculation of properties as function of p,h or of p,s without any iteration, the two
backward equations require extremely good numerical consistency with the basic equation.
The exact requirements for these numerical consistencies were obtained from comprehensive
test calculations for several characteristic power cycles. The result of these investigations,
namely the assignment of the tolerable numerical inconsistencies between the basic equation,
Eq. (7), and the corresponding backward equations, is given in Eqs. (12) and (14),
respectively.
10
5.2.1 The Backward Equation T ( p, h )
The backward equation T ( p,h ) for region 1 has the following dimensionless form:
T ( p, h)
T

20
   ,    ni Ii   1
Ji
(11)
,
i 1
where  = T/T *,  = p/p*, and  = h/h* with T * = 1 K , p* = 1 MPa, and h* = 2500 kJ kg.
The coefficients ni and exponents Ii and Ji of Eq. (11) are listed in Table 6.
Table 6. Numerical values of the coefficients and exponents of the backward equation T ( p,h ) for
region 1, Eq. (11)
i
Ii
Ji
1
0
0
2
0
3
ni
i
Ii
Ji
ni
– 0.238 724 899 245 21 × 103
11
1
4
1
0.404 211 886 379 45 × 103
12
1
10
0.939 654 008 783 63 × 10
0
2
0.113 497 468 817 18 × 103
13
1
32
0.115 736 475 053 40 × 10
4
0
6
– 0.584 576 160 480 39 × 101
14
2
10
– 0.258 586 412 820 73 × 10
5
0
22
– 0.152 854 824 131 40 × 10
15
2
32
– 0.406 443 630 847 99 × 10
6
0
32
– 0.108 667 076 953 77 × 10
16
3
10
0.664 561 861 916 35 × 10
7
1
0
– 0.133 917 448 726 02 × 102
17
3
32
0.806 707 341 030 27 × 10
8
1
1
0.432 110 391 835 59 × 102
18
4
32
– 0.934 777 712 139 47 × 10
9
1
2
– 0.540 100 671 705 06 × 102
19
5
32
0.582 654 420 206 01 × 10
10
1
3
0.305 358 922 039 16 × 102
20
6
32
– 0.150 201 859 535 03 × 10
– 0.659 647 494 236 38 × 101
Range of validity
Equation (11) covers the same range of validity as the basic equation, Eq. (7), except for
the metastable region (superheated liquid), where Eq. (11) is not valid.
Numerical consistency with the basic equation
For ten million random pairs of p and h covering the entire region 1, the differences T
between temperatures from Eq. (11) and from Eq. (7) were calculated and the absolute
maximum difference T  max and the root-mean-square difference TRMS were determined.
These actual inconsistency values and the tolerated value T  tol (see the beginning of
Section 5.2) amount to:
T  tol = 25 mK ; TRMS = 13.4 mK ; T  max = 23.6 mK .
(12)
11
Computer-program verification
To assist the user in computer-program verification of Eq. (11), Table 7 contains the
corresponding test values.
Table 7. Temperature values calculated from Eq. (11) for selected
values of p and h a
p / MPa
h/(kJ kg–1)
T/K
3
500
0.391 798 509 × 10
80
500
0.378 108 626 × 10
80
1500
0.611 041 229 × 10
3
3
3
a It is recommended to verify the programmed equation (8 byte real values)
for all three combinations of p and h given in this table.
5.2.2 The Backward Equation T ( p, s )
The backward equation T ( p,s ) for region 1 has the following dimensionless form:
T ( p, s)
T
20
   ,     ni  Ii   2  i ,
J
(13)
i 1
where  = T/T *,  = p/p*, and  = s/s* with T * = 1 K , p* = 1 MPa, and s* = 1 kJ kg K.
The coefficients ni and exponents Ii and Ji of Eq. (13) are listed in Table 8.
Table 8. Numerical values of the coefficients and exponents of the backward equation T ( p,s ) for
region 1, Eq. (13)
i
Ii
Ji
ni
i
Ii
Ji
ni
1
0
0
0.174 782 680 583 07 × 103
11
1
12
0.356 721 106 073 66 × 10
2
0
1
0.348 069 308 928 73 × 102
12
1
31
0.173 324 969 948 95 × 10
3
0
2
0.652 925 849 784 55 × 101
13
2
0
0.566 089 006 548 37 × 10
4
0
3
0.330 399 817 754 89
14
2
1
– 0.326 354 831 397 17 × 10
5
0
11
– 0.192 813 829 231 96 × 10
15
2
2
0.447 782 866 906 32 × 10
6
0
31
– 0.249 091 972 445 73 × 10
16
2
9
– 0.513 221 569 085 07 × 10
7
1
0
– 0.261 076 364 893 32
17
2
31
– 0.425 226 570 422 07 × 10
8
1
1
0.225 929 659 815 86
18
3
10
0.264 004 413 606 89 × 10
9
1
2
– 0.642 564 633 952 26 × 10
19
3
32
0.781 246 004 597 23 × 10
10
1
3
0.788 762 892 705 26 × 10
20
4
32
– 0.307 321 999 036 68 × 10
12
Range of validity
Equation (13) covers the same range of validity as the basic equation, Eq. (7), except for
the metastable region (superheated liquid), where Eq. (13) is not valid.
Numerical consistency with the basic equation
For ten million random pairs of p and s covering the entire region 1, the differences T
between temperatures from Eq. (13) and from Eq. (7) were calculated and the absolute
maximum difference T  max and the root-mean-square difference TRMS were determined.
These actual differences and the tolerated value T  tol (see the beginning of Section 5.2)
amount to:

T  tol = 25 mK ; TRMS = 5.8 mK ; T  max = 21.8 mK .
(14)
Computer-program verification
To assist the user in computer-program verification of Eq. (13), Table 9 contains the
corresponding test values.
Table 9.
Temperature values calculated from Eq. (13) for selected
values of p and s a
p / MPa
s/(kJ kg–1 K–1)
T/K
3
0.5
0.307 842 258 × 10
80
0.5
0.309 979 785 × 10
80
3
0.565 899 909 × 10
3
3
3
a It is recommended to verify the programmed equation using 8 byte real
values for all three combinations of p and s given in this table.
6 Equations for Region 2
This section contains all details relevant for the use of the basic and backward equations of
region 2 of IAPWS-IF97. Information about the consistency of the basic equation of this
region with the basic equations of regions 3, 4 and 5 along the corresponding region
boundaries is summarized in Section 10. The auxiliary equation for defining the boundary
between regions 2 and 3 is given in Section 4. Section 11 contains the results of computingtime comparisons between IAPWS-IF97 and IFC-67. The estimates of uncertainty of the most
relevant properties can be found in Section 12.
13
6.1 Basic Equation
The basic equation for this region is a fundamental equation for the specific Gibbs free
energy g. This equation is expressed in dimensionless form,  = g/( RT ), and is separated into
two parts, an ideal-gas part  o and a residual part r, so that
g  p, T 
   ,    o   ,    r  ,  ,
RT
(15)
where  = p/p* and  = T */T with R given by Eq. (1).
The equation for the ideal-gas part  o of the dimensionless Gibbs free energy reads
9
o
 o  ln    nio Ji ,
(16)
i 1
where  = p/p* and  = T */T with p* = 1 MPa and T * = 540 K. The coefficients n 1o and n o2
were adjusted in such a way that the values for the specific internal energy and specific
entropy in the ideal-gas state relate to Eq. (8). Table 10 contains the coefficients nio and
exponents Jio of Eq. (16).
Table 10. Numerical values of the coefficients and exponents of the ideal-gas part  o of the
dimensionless Gibbs free energy for region 2, Eq. (16)a
i
Jio
nio
i
Jio
nio
1a
0
– 0.969 276 865 002 17 × 101
6
–2
0.142 408 191 714 44 × 101
2a
1
0.100 866 559 680 18 × 102
7
–1
– 0.438 395 113 194 50 × 101
3
–5
– 0.560 879 112 830 20 × 10
8
2
4
–4

9
3
5
–3
0.714 527 380 814 55 × 10
– 0.284 086 324 607 72
0.212 684 637 533 07 × 10
– 0.407 104 982 239 28
a If Eq. (16) is incorporated into Eq. (18), instead of the values for n o and n o given above, the following values
1
2
for these two coefficients must be used: n 1o = –0.969 372 683 930 49 × 101 , n o2 =0.100 872 759 700 06 × 102.
The form of the residual part  r of the dimensionless Gibbs free energy is as follows:
43
   ni  Ii   0.5 
r
Ji
,
(17)
i 1
where  = p/p* and  = T */T with p* = 1 MPa and T * = 540 K. The coefficients ni and
exponents Ii and Ji of Eq. (17) are listed in Table 11.
14
Table 11. Numerical values of the coefficients and exponents of the
residual part  r of the dimensionless Gibbs free energy for
region 2, Eq. (17)
i
Ii
Ji
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
5
6
6
6
7
7
7
8
8
9
10
10
10
16
16
18
20
20
20
21
22
23
24
24
24
0
1
2
3
6
1
2
4
7
36
0
1
3
6
35
1
2
3
7
3
16
35
0
11
25
8
36
13
4
10
14
29
50
57
20
35
48
21
53
39
26
40
58
ni
– 0.177 317 424 732 13  10
– 0.178 348 622 923 58  10
– 0.459 960 136 963 65  10
– 0.575 812 590 834 32  10
– 0.503 252 787 279 30  10
– 0.330 326 416 702 03  10
– 0.189 489 875 163 15  10
– 0.393 927 772 433 55  10
– 0.437 972 956 505 73  10
– 0.266 745 479 140 87  10
0.204 817 376 923 09  10
0.438 706 672 844 35  10
– 0.322 776 772 385 70  10
– 0.150 339 245 421 48  10
– 0.406 682 535 626 49  10
– 0.788 473 095 593 67  10
0.127 907 178 522 85  10
0.482 253 727 185 07  10
0.229 220 763 376 61  10
– 0.167 147 664 510 61  10
– 0.211 714 723 213 55  10
– 0.238 957 419 341 04  102
– 0.590 595 643 242 70  10
– 0.126 218 088 991 01  10
– 0.389 468 424 357 39  10
0.112 562 113 604 59  10
– 0.823 113 408 979 98  10 1
0.198 097 128 020 88  10
0.104 069 652 101 74  10
– 0.102 347 470 959 29  10
– 0.100 181 793 795 11  10
– 0.808 829 086 469 85  10
0.106 930 318 794 09
– 0.336 622 505 741 71
0.891 858 453 554 21  10
0.306 293 168 762 32  10
– 0.420 024 676 982 08  10
– 0.590 560 296 856 39  10
0.378 269 476 134 57  10
– 0.127 686 089 346 81  10
0.730 876 105 950 61  10
0.554 147 153 507 78  10
– 0.943 697 072 412 10  10
15
All thermodynamic properties can be derived from Eq. (15) by using the appropriate
combinations of the ideal-gas part  o, Eq. (16), and the residual part  r, Eq. (17), of the
dimensionless Gibbs free energy and their derivatives. Relations between the relevant
thermodynamic properties and  o and  r and their derivatives are summarized in Table 12.
All required derivatives of the ideal-gas part and of the residual part of the dimensionless
Gibbs free energy are explicitly given in Table 13 and Table 14, respectively.
Table 12. Relations of thermodynamic properties to the ideal-gas part  o and the residual part  r of
the dimensionless Gibbs free energy and their derivatives a when using Eq. (15) or Eq. (18)
Property
Relation
Specific volume
v   g/ p T
v( , )
Specific internal energy
u  ,  
u  g  T  g/T  p  p (g/  p)T
Specific entropy
RT
s  ,  
s    g/T  p
R
h  ,  
Specific enthalpy
h  g  T  g/T  p
RT
Specific isobaric heat capacity
c p   h /T  p

p
   o   r
RT
cv    u /T v
1/2
w  v     p /  v s 

 


   o   r
cv  , 

  2  o   r



r 2
(1   r   
)
r
1   2 
1  2 r   2 r 2
1     
1           
   
r
r
2 r
2
a


  2  o   r 
w2 ( , )

RT
Speed of sound
 
   o   r   o   r
c p  , 
R

   o   r    o   r
R
Specific isochoric heat capacity

o
r
  r 
  2 r 
  r 
  2 r 
  2 r 
r
r
r
r
 ,     2  ,    
 ,     2  ,    
,
  
  
  
  
   
 r  
  o 
  2 o 
o
 ,     2 
  
  
 o  
2
16
Table 13. The ideal-gas part  o of the dimensionless Gibbs
free energy and its derivatives a according to
Eq. (16)

ln 
=
o
9
+
 nio J
o
i
i 1
1/
+
0
=  1/ 2
+
0
=
+
 nio Jio J
 o
=
o
 
9
o

0
o
i 1
i 1
9
 
=
o
 
=
o
a
0
+
 nio Jio  Jio  1 J
o
i 2
i 1
0
+
0
  o 
  2 o 
  o 
  2 o 
  2 o 
o
o
o
o
,
,
,
,








   2 


 2 





  
  
  
  
   
 o  
Table 14. The residual part r of the dimensionless Gibbs free energy and its derivatives a according
to Eq. (17)
43
 r   ni  Ii   0.5 
Ji
i 1
43
    ni I i 
r
i 1
 r 
Ii 1
  0.5
43
 ni  Ii J i   0.5
Ji
Ji
i 1
J i 1
i 1
43
 r 
    ni I i  Ii 1 J i   0.5 
r
43
    ni I i  I i  1  Ii 2   0.5 
r
43
 ni  I J i  J i  1   0.5
i
J i 2
i 1
J i 1
i 1
a
  r 
  2 r 
  r 
  2 r 
  2 r 
r
r
r
r
 ,     2  ,    
 ,     2  ,    

  
  
  
  
   
 r  
17
Range of validity
Equation (15) covers region 2 of IAPWS-IF97 defined by the following range of
temperature and pressure, see Fig. 1:
273.15 K  T  623.15 K
623.15 K  T  863.15 K
863.15 K  T  1073.15 K
0  p  ps ( T )Eq.(30)
0  p  p ( T )Eq.(5)
0  p  100 MPa
In addition to the properties in the stable single-phase vapor region, Eq. (15) also yields
reasonable values in the metastable-vapor region for pressures above 10 MPa. Equation (15)
is not valid in the metastable-vapor region at pressures p  10 MPa; for this part of the
metastable-vapor region see Section 6.2.
Note: For temperatures between 273.15 K and 273.16 K, the part of the range of validity between the pressures
on the saturation-pressure line, Eq. (30), and on the sublimation line [10] corresponds to metastable
states.
Computer-program verification
To assist the user in computer-program verification of Eq. (15), Table 15 contains test
values of the most relevant properties.
Table 15. Thermodynamic property values calculated from Eq. (15) for selected values of T and p a
T = 300 K,
p = 0.0035 MPa
T = 700 K,
p = 0.0035 MPa
T = 700 K,
p = 30 MPa
v / (m3 kg–1)
0.394 913 866 × 102
0.923 015 898 × 102
0.542 946 619 × 10–2
h / (kJ kg–1)
0.254 991 145 × 104
0.333 568 375 × 104
0.263 149 474 × 104
u / (kJ kg–1)
0.241 169 160 × 104
0.301 262 819 × 104
0.246 861 076 × 104
s / (kJ kg–1 K–1)
0.852 238 967 × 101
0.101 749 996 × 102
0.517 540 298 × 101
cp / (kJ kg–1 K–1)
0.191 300 162 × 101
0.208 141 274 × 101
0.103 505 092 × 102
w / (m s–1)
0.427 920 172 × 103
0.644 289 068 × 103
0.480 386 523 × 103
a It is recommended to verify programmed functions using 8 byte real values for all three combinations of T and
p given in this table.
6.2 Supplementary Equation for the Metastable-Vapor Region
As for the basic equation, Eq. (15), the supplementary equation for a part of the
metastable-vapor region bounding region 2 is given in the dimensionless form of the specific
Gibbs free energy,  = g/( RT ), consisting of an ideal-gas part  o and a residual part r, so that
g  p, T 
RT
    ,     o  ,     r   ,   ,
where  = p/p* and  = T */T with R given by Eq. (1).
(18)
18
The equation for the ideal-gas part  o is identical with Eq. (16) except for the values of the
two coefficients n 1o and n o2 , see Table 10. For the use of Eq. (16) as part of Eq. (18) the
coefficients n 1o and n o2 were slightly readjusted to meet the high consistency requirement
between Eqs. (18) and (15) regarding the properties h and s along the saturated vapor line; see
below.
The equation for the residual part r reads
13
 r   ni  Ii   0.5 
Ji
,
(19)
i 1
where  = p/p* and  = T */T with p* = 1 MPa and T * = 540 K. The coefficients ni and
exponents Ii and Ji of Eq. (19) are listed in Table 16.
Note: In the metastable-vapor region there are no experimental data to which an equation can be fitted. Thus,
Eq. (18) is only based on input values extrapolated from the stable single-phase region 2. These
extrapolations were performed with a special low-density gas equation [11] considered to be more
suitable for such extrapolations into the metastable-vapor region than IAPWS-95 [3].
Table 16. Numerical values of the coefficients and exponents of the
residual part  r of the dimensionless Gibbs free energy for
the metastable-vapor region, Eq. (19)
i
Ii
Ji
ni
1
1
0
– 0.733 622 601 865 06  10
2
1
2
– 0.882 238 319 431 46  10
3
1
5
– 0.723 345 552 132 45  10
4
1
11
– 0.408 131 785 344 55  10
5
2
1
0.200 978 033 802 07  10
6
2
7
– 0.530 459 218 986 42  10
7
2
16
– 0.761 904 090 869 70  10
8
3
4
– 0.634 980 376 573 13  10
9
3
16
– 0.860 430 930 285 88  10
10
4
7
0.753 215 815 227 70  10
11
4
10
– 0.792 383 754 461 39  10
12
5
9
– 0.228 881 607 784 47  10
13
5
10
– 0.264 565 014 828 10  10
All thermodynamic properties can be derived from Eq. (18) by using the appropriate
combinations of the ideal-gas part  o, Eq. (16), and the residual part  r, Eq. (19), of the
dimensionless Gibbs free energy and their derivatives. Relations between the relevant
thermodynamic properties and  o and  r and their derivatives are summarized in Table 12.
19
All required derivatives of the ideal-gas part and of the residual part of the dimensionless
Gibbs free energy are explicitly given in Table 13 and Table 17, respectively.
Table 17. The residual part r of the dimensionless Gibbs free energy and its derivatives a according
to Eq. (19)
13
 r   ni  Ii   0.5 
Ji
i 1
13
 r   ni I i  Ii 1   0.5 
Ji
i 1
 r 
13
 ni  Ii J i   0.5
13
r
 
  ni I i  I i  1  Ii 2   0.5 
Ji
i 1
J i 1
i 1
 r 
13
r
 
  ni I i  Ii 1 J i   0.5 
13
 ni  I J i  J i  1   0.5
i
J i 2
i 1
J i 1
i 1
a
  r 
  2 r 
  r 
  2 r 
  2 r 
r
r
r
r
,
,
,
,









 2


 2 






  
  
  
  
   
 r  
Range of validity
Equation (18) is valid in the metastable-vapor region from the saturated-vapor line to the
5 % equilibrium moisture line (determined from the equilibrium h and h values calculated at
the given pressure) for pressures from the triple-point pressure, see Eq. (9), up to 10 MPa.
Consistency with the basic equation
The consistency of Eq. (18) with the basic equation, Eq. (15), along the saturated vapor line
is characterized by the following maximum inconsistencies regarding the properties v, h, cp ,
s, g, and w :
v max = 0.014 %
s max = 0.082 J kg1 K1
h max = 0.043 kJ kg1 g max = 0.023 kJ kg1
cp  max = 0.78 %
w max = 0.051 % .
These maximum inconsistencies are clearly smaller than the consistency requirements on
region boundaries corresponding to the so-called Prague values [13], which are given in
Section 10.
20
Along the 10 MPa isobar in the metastable-vapor region, the transition between Eq. (18)
and Eq. (15) is not smooth, which is however, not of importance for practical calculations.
Computer-program verification
To assist the user in computer-program verification of Eq. (18), Table 18 contains test
values of the most relevant properties.
Table 18. Thermodynamic property values calculated from Eq. (18) for selected values of T and p a
T = 450 K,
p = 1 MPa
v / (m3 kg–1)
–1
T = 440 K,
p = 1 MPa
0.192 516 540
T = 450 K,
p = 1.5 MPa
0.186 212 297
4
0.274 015 123 × 10
0.121 685 206
4
0.272 134 539 × 104
h / (kJ kg )
0.276 881 115 × 10
u / (kJ kg–1)
0.257 629 461 × 104
0.255 393 894 × 104
0.253 881 758 × 104
s / (kJ kg–1 K–1)
0.656 660 377 × 101
0.650 218 759 × 101
0.629 170 440 × 101
cp / (kJ kg–1 K–1)
0.276 349 265 × 101
0.298 166 443 × 101
0.362 795 578 × 101
w / (m s–1)
0.498 408 101 × 103
0.489 363 295 × 103
0.481 941 819 × 103
a It is recommended to verify programmed functions using 8 byte real values for all three combinations of T and
p given in this table.
6.3 Backward Equations
For the calculation of properties as function of p, h or of p, s without any iteration, the two
backward equations require extremely good numerical consistency with the basic equation.
The exact requirements for these numerical consistencies were obtained from comprehensive
test calculations for several characteristic power cycles. The result of these investigations,
namely the assignment of the tolerable numerical inconsistencies between the basic equation,
Eq. (15), and the corresponding backward equations, is given in Tables 23 and 28,
respectively.
Region 2 is covered by three T ( p, h ) and three T ( p, s ) equations. Figure 2 shows the way
in which region 2 is divided into three subregions for the backward equations. The boundary
between the subregions 2a and 2b is the isobar p = 4 MPa; the boundary between the
subregions 2b and 2c corresponds to the entropy line s = 5.85 kJ kg1 K1.
21
Fig. 2. Division of region 2 of IAPWS-IF97 into the three subregions 2a,
2b, and 2c for the backward equations T( p,h ) and T( p,s ).
In order to know whether the T( p,h ) equation for subregion 2b or for subregion 2c has to
be used for given values of p and h, a special correlation equation for the boundary between
subregions 2b and 2c (which approximates s = 5.85 kJ kg1 K1) is needed; see Fig. 2. This
boundary equation, called the B2bc-equation, is a simple quadratic pressure-enthalpy relation
which reads
  n1  n2  n3 2 ,
(20)
where  = p/p* and  = h/h* with p* = 1 MPa and h* = 1 kJ kg1. The coefficients n1 to n3 of
Eq. (20) are listed in Table 19. Based on its simple form, Eq. (20) does not describe exactly
the isentropic line s = 5.85 kJ kg K; the entropy values corresponding to this p-h relation
are between s = 5.81 kJ kg1 K1 and s = 5.85 kJ kg1 K1. The enthalpy-explicit form of
Eq. (20) is as follows:
1/2
  n4    n5  / n3 
,
(21)
with  and  according to Eq. (20) and the coefficients n3 to n5 listed in Table 19. Equations
(20) and (21) give the boundary line between subregions 2b and 2c from the saturation state at
T = 554.485 K and ps = 6.546 70 MPa to T = 1019.32 K and p = 100 MPa.
For the backward equations T( p,s ) the boundary between subregions 2b and 2c is, based
on the value s = 5.85 kJ kg1 K1 along this boundary, automatically defined for given values
of p and s.
22
Table 19. Numerical values of the coefficients of the B2bc-equation, Eqs. (20) and (21),
for defining the boundary between subregions 2b and 2c with respect to
T( p,h ) calculations
i
ni
1
0.905 842 785 147 23 × 10
2
3
– 0.679 557 863 992 41
i
ni
4
0.265 265 719 084 28 × 104
5
0.452 575 789 059 48 × 101
0.128 090 027 301 36 × 10
3
For computer-program verification, Eqs. (20) and (21) must meet the following p-h point:
p = 0.100 000 000  103 MPa , h = 0.351 600 432 3  104 kJ kg1.
6.3.1 The Backward Equations T( p, h ) for Subregions 2a, 2b, and 2c
The backward equation T( p,h ) for subregion 2a in its dimensionless form reads
T2a ( p, h)
T

34
 2a  ,    ni  Ii   2.1
Ji
(22)
,
i 1
where  = T/T * ,  = p/p* , and  = h/h* with T * = 1 K , p* = 1 MPa, and h* = 2000 kJ kg1.
The coefficients ni and exponents Ii and Ji of Eq. (22) are listed in Table 20.
Table 20. Numerical values of the coefficients and exponents of the backward equation T ( p,h ) for
subregion 2a, Eq. (22)
i
Ii
Ji
ni
i
Ii
Ji
ni
1
0
0
0.108 989 523 182 88 × 104
18
2
7
0.116 708 730 771 07 × 102
2
0
1
0.849 516 544 955 35 × 103
19
2
36
0.128 127 984 040 46 × 109
3
0
2
– 0.107 817 480 918 26 × 103
20
2
38
– 0.985 549 096 232 76 × 109
4
0
3
0.331 536 548 012 63 × 102
21
2
40
0.282 245 469 730 02 × 1010
5
0
7
– 0.742 320 167 902 48 × 101
22
2
42
– 0.359 489 714 107 03 × 1010
6
0
20
0.117 650 487 243 56 × 102
23
2
44
0.172 273 499 131 97 × 1010
7
1
0
0.184 457 493 557 90 × 101
24
3
24
– 0.135 513 342 407 75 × 105
8
1
1
– 0.417 927 005 496 24 × 101
25
3
44
0.128 487 346 646 50 × 108
9
1
2
0.624 781 969 358 12 × 101
26
4
12
0.138 657 242 832 26 × 101
10
1
3
– 0.173 445 631 081 14 × 102
27
4
32
0.235 988 325 565 14 × 106
11
1
7
– 0.200 581 768 620 96 × 103
28
4
44
– 0.131 052 365 450 54 × 108
12
1
9
0.271 960 654 737 96 × 103
29
5
32
0.739 998 354 747 66 × 104
13
1
11
– 0.455 113 182 858 18 × 103
30
5
36
– 0.551 966 970 300 60 × 106
14
1
18
0.309 196 886 047 55 × 104
31
5
42
0.371 540 859 962 33 × 107
15
1
44
0.252 266 403 578 72 × 106
32
6
34
0.191 277 292 396 60 × 105
16
2
0
– 0.617 074 228 683 39 × 10
33
6
44
– 0.415 351 648 356 34 × 106
17
2
2
– 0.310 780 466 295 83
34
7
28
– 0.624 598 551 925 07 × 102
23
The backward equation T( p,h ) for subregion 2b in its dimensionless form reads
T2b ( p, h)
T

38
 2 b  ,    ni   2  i   2.6 
I
Ji
(23)
,
i 1
where  = T/T *,  = p/p* , and  = h/h* with T * = 1 K , p* = 1 MPa, and h* = 2000 kJ kg1.
The coefficients ni and exponents Ii and Ji of Eq. (23) are listed in Table 21.
Table 21. Numerical values of the coefficients and exponents of the backward equation T ( p,h ) for
subregion 2b, Eq. (23)
i
Ii
Ji
1
0
0
2
0
3
ni
i
Ii
Ji
0.148 950 410 795 16 × 104
20
2
40
0.712 803 519 595 51 × 10
1
0.743 077 983 140 34 × 103
21
3
1
0.110 328 317 899 99 × 10
0
2
– 0.977 083 187 978 37 × 102
22
3
2
0.189 552 483 879 02 × 10
4
0
12
0.247 424 647 056 74 × 101
23
3
12
0.308 915 411 605 37 × 10
5
0
18
24
3
24
0.135 555 045 549 49 × 10
6
0
24
25
4
2
0.286 402 374 774 56 × 10
7
0
28
26
4
12
– 0.107 798 573 575 12 × 10
8
0
40
0.852 081 234 315 44 × 10
27
4
18
– 0.764 627 124 548 14 × 10
9
1
0
0.937 471 473 779 32
28
4
24
0.140 523 928 183 16 × 10
10
1
2
0.335 931 186 049 16 × 101
29
4
28
– 0.310 838 143 314 34 × 10
11
1
6
0.338 093 556 014 54 × 101
30
4
40
– 0.103 027 382 121 03 × 10
12
1
12
0.168 445 396 719 04
31
5
18
0.282 172 816 350 40 × 10
13
1
18
0.738 757 452 366 95
32
5
24
0.127 049 022 719 45 × 10
14
1
24
– 0.471 287 374 361 86
33
5
40
0.738 033 534 682 92 × 10
15
1
28
0.150 202 731 397 07
34
6
28
– 0.110 301 392 389 09 × 10
16
1
40
– 0.217 641 142 197 50 × 10
35
7
2
– 0.814 563 652 078 33 × 10
17
2
2
– 0.218 107 553 247 61 × 10
36
7
28
– 0.251 805 456 829 62 × 10
18
2
8
– 0.108 297 844 036 77
37
9
1
– 0.175 652 339 694 07 × 10
19
2
18
– 0.463 333 246 358 12 × 10
38
9
40
0.869 341 563 441 63 × 10
– 0.632 813 200 160 26
0.113 859 521 296 58 × 101
– 0.478 118 636 486 25
ni
The backward equation T( p,h ) for subregion 2c in its dimensionless form reads
T2c ( p, h)
T
23
 2c  ,    ni   25  i   1.8 
I
Ji
,
(24)
i 1
where  = T/T *,  = p/p*, and  = h/h* with T * = 1 K , p* = 1 MPa, and h* = 2000 kJ kg1.
The coefficients ni and exponents Ii and Ji of Eq. (24) are listed in Table 22.
24
Table 22. Numerical values of the coefficients and exponents of the backward
equation T ( p,h ) for subregion 2c, Eq. (24)
i
Ii
Ji
ni
1
–7
0
– 0.323 683 985 552 42 × 1013
2
–7
4
0.732 633 509 021 81 × 1013
3
–6
0
0.358 250 899 454 47 × 1012
4
–6
2
– 0.583 401 318 515 90 × 1012
5
–5
0
– 0.107 830 682 174 70 × 1011
6
–5
2
0.208 255 445 631 71 × 1011
7
–2
0
0.610 747 835 645 16 × 106
8
–2
1
0.859 777 225 355 80 × 106
9
–1
0
– 0.257 457 236 041 70 × 105
10
–1
2
0.310 810 884 227 14 × 105
11
0
0
0.120 823 158 659 36 × 104
12
0
1
0.482 197 551 092 55 × 103
13
1
4
0.379 660 012 724 86 × 101
14
1
8
– 0.108 429 848 800 77 × 102
15
2
4
– 0.453 641 726 766 60 × 10
16
6
0
0.145 591 156 586 98 × 10
17
6
1
0.112 615 974 072 30 × 10
18
6
4
– 0.178 049 822 406 86 × 10
19
6
10
0.123 245 796 908 32 × 10
20
6
12
– 0.116 069 211 309 84 × 10
21
6
16
0.278 463 670 885 54 × 10
22
6
20
– 0.592 700 384 741 76 × 10
23
6
22
0.129 185 829 918 78 × 10
Range of validity
Equations (22), (23), and (24) are only valid in the respective subregion 2a, 2b, and 2c
which do not include the metastable-vapor region. The boundaries between these subregions
are defined at the beginning of Section 6.3; the lowest pressure for which Eq. (22) is valid
amounts to 611.2127 Pa corresponding to the saturation pressure at 273.15 K calculated from
Eq. (30).
Numerical consistency with the basic equation
For ten million random pairs of p and h covering each of the subregions 2a, 2b, and 2c, the
differences T between temperatures calculated from Eqs. (22) to (24), respectively, and from
Eq. (15) were determined. The corresponding maximum and root-mean-square differences are
25
listed in Table 23 together with the tolerated differences according to the numerical
consistency requirements with respect to Eq. (15).
Table 23. Maximum differences  T  max and root-mean-square differences TRMS between
temperatures calculated from Eqs. (22) to (24), and from Eq. (15) in comparison with the
tolerated differences  T  tol
Subregion
Equation
T  tol /mK
T  max /mK
TRMS /mK
2a
2b
2c
22
23
24
10
10
25
9.3
9.6
23.7
2.9
3.9
10.4
Computer-program verification
To assist the user in computer-program verification of Eqs. (22) to (24), Table 24 contains
the corresponding test values.
Table 24. Temperature values calculated from Eqs. (22) to (24) for
selected values of p and h a
Equation
p / MPa
h/(kJ kg1)
T/K
0.001
3000
0.534 433 241 × 103
3
3000
0.575 373 370 × 103
3
4000
0.101 077 577 × 104
5
3500
0.801 299 102 × 103
5
4000
0.101 531 583 × 104
25
3500
0.875 279 054 × 103
40
2700
0.743 056 411 × 103
60
2700
0.791 137 067 × 103
60
3200
0.882 756 860 × 103
22
23
24
a It is recommended to verify the programmed equations using 8 byte real values
for all three combinations of p and h given in this table for each of the
equations.
6.3.2 The Backward Equations T( p, s ) for Subregions 2a, 2b, and 2c
The backward equation T( p,s ) for subregion 2a in its dimensionless form reads
T2a ( p, s)
T

46
 2a  ,     ni  Ii   2 
Ji
,
(25)
i 1
where  = T/T *,  = p/p*, and  = s/s* with T * = 1 K , p* = 1 MPa, and s* = 2 kJ kg1 K1.
The coefficients ni and exponents Ii and Ji of Eq. (25) are listed in Table 25.
26
Table 25. Numerical values of the coefficients and exponents of the backward equation T( p,s ) for
subregion 2a, Eq. (25)
i
Ii
Ji
ni
i
1
–1.5
–24
– 0.392 359 838 619 84  106
24
–0.25
–11
– 0.597 806 388 727 18  104
2
–1.5
–23
0.515 265 738 272 70  106
25
–0.25
–6
– 0.704 014 639 268 62  103
3
–1.5
–19
0.404 824 431 610 48  105
26
0.25
1
0.338 367 841 075 53  103
4
–1.5
–13
– 0.321 937 909 239 02  103
27
0.25
4
0.208 627 866 351 87  10
5
–1.5
–11
0.969 614 242 186 94  102
28
0.25
8
0.338 341 726 561 96  10
6
–1.5
–10
– 0.228 678 463 717 73  102
29
0.25
11
– 0.431 244 284 148 93  10
7
–1.25
–19
– 0.449 429 141 243 57  106
30
0.5
0
0.166 537 913 564 12  103
8
–1.25
–15
– 0.501 183 360 201 66  104
31
0.5
1
– 0.139 862 920 558 98  103
9
–1.25
–6
– 0.788 495 479 998 72
0.356 844 635 600 15
Ii
Ji
ni
32
0.5
5
5
33
0.5
6
0.721 324 117 538 72  10
10
–1.0
–26
0.442 353 358 481 90  10
11
–1.0
–21
– 0.136 733 888 117 08  105
34
0.5
10
– 0.597 548 393 982 83  10
12
–1.0
–17
0.421 632 602 078 64  106
35
0.5
14
– 0.121 413 589 539 04  10
13
–1.0
–16
0.225 169 258 374 75  105
36
0.5
16
0.232 270 967 338 71  10
14
–1.0
–9
0.474 421 448 656 46  103
37
0.75
0
– 0.105 384 635 661 94  10
15
–1.0
–8
– 0.149 311 307 976 47  103
38
0.75
4
0.207 189 254 965 02  10
16
–0.75
–15
– 0.197 811 263 204 52  106
39
0.75
9
– 0.721 931 552 604 27  10
17
–0.75
–14
– 0.235 543 994 707 60  105
40
0.75
17
0.207 498 870 811 20  10
18
–0.5
–26
– 0.190 706 163 020 76  105
41
1.0
7
– 0.183 406 579 113 79  10
19
–0.5
–13
0.553 756 698 831 64  105
42
1.0
18
0.290 362 723 486 96  10
20
–0.5
–9
0.382 936 914 373 63  104
43
1.25
3
21
–0.5
–7
– 0.603 918 605 805 67  103
44
1.25
15
0.256 812 397 299 99  10
22
–0.25
–27
0.193 631 026 203 31  104
45
1.5
5
– 0.127 990 029 337 81  10
23
–0.25
–25
0.426 606 436 986 10  104
46
1.5
18
– 0.821 981 026 520 18  10
0.210 375 278 936 19
The backward equation T( p,s ) for subregion 2b in its dimensionless form reads
T2b ( p, s)
T

44
 2b  ,     ni  Ii 10   
Ji
,
(26)
i 1
where  = T/T *,  = p/p*, and  = s/s* with T * = 1 K , p* = 1 MPa, and s* = 0.7853 kJ kg1 K1.
The coefficients ni and exponents Ii and Ji of Eq. (26) are listed in Table 26.
27
Table 26. Numerical values of the coefficients and exponents of the backward equation T( p,s ) for
subregion 2b, Eq. (26)
i
Ii
Ji
ni
i
Ii
Ji
ni
1
–6
0
0.316 876 650 834 97  106
23
0
2
0.417 273 471 596 10  102
2
–6
11
0.208 641 758 818 58  102
24
0
4
0.219 325 494 345 32  101
3
–5
0
– 0.398 593 998 035 99  106
25
0
5
– 0.103 200 500 090 77  101
4
–5
11
– 0.218 160 585 188 77  102
26
0
6
0.358 829 435 167 03
5
–4
0
0.223 697 851 942 42  106
27
0
9
0.525 114 537 260 66  10
6
–4
1
– 0.278 417 034 458 17  104
28
1
0
0.128 389 164 507 05  102
7
–4
11
0.992 074 360 714 80  101
29
1
1
– 0.286 424 372 193 81  101
8
–3
0
– 0.751 975 122 991 57  105
30
1
2
9
–3
1
0.297 086 059 511 58  104
31
1
3
– 0.999 629 545 849 31  10
10
–3
11
– 0.344 068 785 485 26  101
32
1
7
– 0.326 320 377 784 59  10
11
–3
12
0.388 155 642 491 15
33
1
8
0.233 209 225 767 23  10
12
–2
0
0.175 112 950 857 50  105
34
2
0
13
–2
1
– 0.142 371 128 544 49  104
35
2
1
0.290 722 882 399 02  10
14
–2
6
0.109 438 033 641 67  101
36
2
5
0.375 347 027 411 67  10
15
–2
10
0.899 716 193 084 95
37
3
0
0.172 966 917 024 11  10
16
–1
0
– 0.337 597 400 989 58  104
38
3
1
– 0.385 560 508 445 04  10
17
–1
1
0.471 628 858 183 55  103
39
3
3
– 0.350 177 122 926 08  10
18
–1
5
– 0.191 882 419 936 79  101
40
4
0
– 0.145 663 936 314 92  10
19
–1
8
0.410 785 804 921 96
41
4
1
0.564 208 572 672 69  10
20
–1
9
– 0.334 653 781 720 97
42
5
0
0.412 861 500 746 05  10
21
0
0
0.138 700 347 775 05  104
43
5
1
– 0.206 846 711 188 24  10
22
0
1
– 0.406 633 261 958 38  103
44
5
2
0.164 093 936 747 25  10
0.569 126 836 648 55
– 0.153 348 098 574 50
The backward equation T( p,s ) for subregion 2c in its dimensionless form reads
T2c ( p, s)
T

30
 2c  ,     ni  Ii  2   
Ji
,
(27)
i 1
where  = T/T *,  = p/p*, and  = s/s* with T * = 1 K , p* = 1 MPa, and s* = 2.9251 kJ kg1 K1.
The coefficients ni and exponents Ii and Ji of Eq. (27) are listed in Table 27.
28
Table 27. Numerical values of the coefficient s and exponents of the backward equation T( p,s ) for
subregion 2c, Eq. (27)
i
Ii
Ji
1
–2
0
2
–2
3
ni
i
Ii
Ji
ni
0.909 685 010 053 65  103
16
3
1
– 0.145 970 082 847 53  10
1
0.240 456 670 884 20  104
17
3
5
0.566 311 756 310 27  10
–1
0
– 0.591 623 263 871 30  103
18
4
0
– 0.761 558 645 845 77  10
4
0
0
0.541 454 041 280 74  103
19
4
1
0.224 403 429 193 32  10
5
0
1
– 0.270 983 084 111 92  103
20
4
4
– 0.125 610 950 134 13  10
6
0
2
0.979 765 250 979 26  103
21
5
0
0.633 231 326 609 34  10
7
0
3
– 0.469 667 729 594 35  103
22
5
1
– 0.205 419 896 753 75  10
8
1
0
0.143 992 746 047 23  102
23
5
2
0.364 053 703 900 82  10
9
1
1
– 0.191 042 042 304 29  102
24
6
0
– 0.297 598 977 892 15  10
10
1
3
0.532 991 671 119 71  101
25
6
1
0.101 366 185 297 63  10
11
1
4
– 0.212 529 753 759 34  102
26
7
0
0.599 257 196 923 51  10
12
2
0
– 0.311 473 344 137 60
27
7
1
– 0.206 778 701 051 64  10
13
2
1
0.603 348 408 946 23
28
7
3
– 0.208 742 781 818 86  10
14
2
2
– 0.427 648 397 025 09  10
29
7
4
0.101 621 668 250 89  10
15
3
0
0.581 855 972 552 59  10
30
7
5
– 0.164 298 282 813 47  10
Range of validity
Equations (25), (26), and (27) are only valid in the respective subregion 2a, 2b, and 2c
which do not include the metastable-vapor region. The boundaries between these subregions
are defined at the beginning of Section 6.3; the lowest pressure for which Eq. (25) is valid
amounts to 611.153 Pa corresponding to the sublimation pressure [10] at 273.15 K.
Numerical consistency with the basic equation
For ten million random pairs of p and s covering each of the subregions 2a, 2b, and 2c, the
differences T between temperatures calculated from Eqs. (25) to (27), respectively, and from
Eq. (15) were determined. The corresponding maximum and root-mean-square differences are
listed in Table 28 together with the tolerated differences according to the numerical
consistency requirements with respect to Eq. (15).
29
Table 28. Maximum differences  T  max and root-mean-square differences TRMS between
temperatures calculated from Eqs. (25) to (27), and from Eq. (15) in comparison with the
tolerated differences  T  tol
Subregion
Equation
T  tol /mK
T  max /mK
TRMS /mK
2a
2b
2c
25
26
27
10
10
25
8.8
6.5
19.0
1.2
2.8
8.3
Computer-program verification
To assist the user in computer-program verification of Eqs. (25) to (27), Table 29 contains
the corresponding test values.
Table 29. Temperature values calculated from Eqs. (25) to (27) for selected
values of p and s a
Equation
25
26
27
p / MPa
s/(kJ kg–1 K–1)
0.1
7.5
0.399 517 097 × 10
0.1
8
0.514 127 081 × 10
2.5
8
0.103 984 917 × 10
8
6
0.600 484 040 × 10
8
7.5
0.106 495 556 × 10
90
6
0.103 801 126 × 10
20
5.75
0.697 992 849 × 10
80
5.25
0.854 011 484 × 10
80
5.75
0.949 017 998 × 10
T/K
3
3
4
3
4
4
3
3
3
a It is recommended to verify the programmed equations using 8 byte real values for all
three combinations of p and s given in this table for each of the equations.
7 Basic Equation for Region 3
This section contains all details relevant for the use of the basic equation of region 3 of
IAPWS-IF97. Information about the consistency of the basic equation of this region with the
basic equations of regions 1, 2, and 4 along the corresponding region boundaries is
summarized in Section 10. The auxiliary equation for defining the boundary between
regions 2 and 3 is given in Section 4. Section 11 contains the results of computing-time
30
comparisons between IAPWS-IF97 and IFC-67. The estimates of uncertainty of the most
relevant properties can be found in Section 12.
The basic equation for this region is a fundamental equation for the specific Helmholtz
free energy f. This equation is expressed in dimensionless form,   f / ( R T ) , and reads
40
f (, T )
   ,   n1 ln    ni  Ii Ji ,
RT
i 2
(28)
where  =  / * ,  = T */T with  * = c , T * = Tc and R, Tc , and c given by Eqs. (1), (2),
and (4). The coefficients ni and exponents Ii and Ji of Eq. (28) are listed in Table 30.
In addition to representing the thermodynamic properties in the single-phase region,
Eq. (28) meets the phase-equilibrium condition (equality of specific Gibbs free energy for
coexisting vapor + liquid states; see Table 31) along the saturation line for T  623.15 K to Tc.
Moreover, Eq. (28) reproduces exactly the critical parameters according to Eqs. (2) to (4) and
yields zero for the first two pressure derivatives with respect to density at the critical point.
Table 30. Numerical values of the coefficients and exponents of the dimensionless Helmholtz free
energy for region 3, Eq. (28)
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Ii
–
0
0
0
0
0
0
0
1
1
1
1
2
2
2
2
2
2
3
3
Ji
–
0
1
2
7
10
12
23
2
6
15
17
0
2
6
7
22
26
0
2
ni
1
0.106 580 700 285 13  10
– 0.157 328 452 902 39  102
0.209 443 969 743 07  102
– 0.768 677 078 787 16  101
0.261 859 477 879 54  101
– 0.280 807 811 486 20  101
0.120 533 696 965 17  101
– 0.845 668 128 125 02  10
– 0.126 543 154 777 14  101
– 0.115 244 078 066 81  101
0.885 210 439 843 18
– 0.642 077 651 816 07
0.384 934 601 866 71
– 0.852 147 088 242 06
0.489 722 815 418 77  101
– 0.305 026 172 569 65  101
0.394 205 368 791 54  10
0.125 584 084 243 08
– 0.279 993 296 987 10
0.138 997 995 694 60  101
i
Ii
Ji
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
3
3
3
4
4
4
4
5
5
5
6
6
6
7
8
9
9
10
10
11
4
16
26
0
2
4
26
1
3
26
0
2
26
2
26
2
26
0
1
26
ni
– 0.201 899 150 235 70  101
– 0.821 476 371 739 63  10
– 0.475 960 357 349 23
0.439 840 744 735 00  10
– 0.444 764 354 287 39
0.905 720 707 197 33
0.705 224 500 879 67
0.107 705 126 263 32
– 0.329 136 232 589 54
– 0.508 710 620 411 58
– 0.221 754 008 730 96  10
0.942 607 516 650 92  10
0.164 362 784 479 61
– 0.135 033 722 413 48  10
– 0.148 343 453 524 72  10
0.579 229 536 280 84  10
0.323 089 047 037 11  10
0.809 648 029 962 15  10
– 0.165 576 797 950 37  10
– 0.449 238 990 618 15  10
All thermodynamic properties can be derived from Eq. (28) by using the appropriate
combinations of the dimensionless Helmholtz free energy and its derivatives. Relations
31
between the relevant thermodynamic properties and  and its derivatives are summarized in
Table 31. All required derivatives of the dimensionless Helmholtz free energy are explicitly
given in Table 32.
Table 31. Relations of thermodynamic properties to the dimensionless Helmholtz
free energy  and its derivatives a when using Eq. (28)
Property
Relation
Pressure
p   2 ( f /  )T
p  ,  
 
 RT
Specific internal energy
u  f  T   f /T  
u  , 
  
RT
Specific entropy
s  , 
s     f /T  
Specific enthalpy
h  f  T   f /T       f / T
Specific isochoric heat capacity
cv   u /T  
Specific isobaric heat capacity
c p   h /T  p
Speed of sound
w    p/ s
1/2
Phase-equilibrium condition
(Maxwell criterion)
a
R
h  , 
RT
   
   
cv  , 
R
c p  , 
R
w 2  ,  
RT
  2
  
2
2
   


2   2
 2   
2
2
   


 2
ps
ps
     ,  ;
     , 
RT  
RT  
ps  1 1 
      ,     , 
RT      
  2 
  2 
  2 
  
  
    ,    2  ,     ,    2  ,   

  
  
  
  
   
32
Table 32. The dimensionless Helmholtz free energy equation and its derivatives a
according to Eq. (28)
40
  n1 ln    ni  Ii  Ji
i 2
40
40
  n1 /  2   ni Ii  Ii 1  Ii 2  Ji
  n1 /    ni I i  Ii 1  Ji
i 2
i 2
40
40
  0   ni  Ii J i  Ji 1  Ji 2
  0   ni  Ii Ji  Ji 1
i 2
i 2
40
  0   ni I i  Ii 1 J i  Ji 1
i 2
a
  2 
  2 
  2 
  
  
    ,    2  ,     ,    2  ,   

  
  
  
  
   
Range of validity
Equation (28) covers region 3 of IAPWS-IF97 defined by the following range of
temperature and pressure, see Fig. 1:
623.15 K  T  T ( p )Eq.(6)
p ( T )Eq.(5)  p  100 MPa .
In addition to the properties in the stable single-phase region defined above, Eq. (28) also
yields reasonable values in the metastable regions (superheated liquid and subcooled steam)
close to the saturated liquid and saturated vapor line.
Computer-program verification
To assist the user in computer-program verification of Eq. (28), Table 33 contains test
values of the most relevant properties.
Table 33. Thermodynamic property values calculated from Eq. (28) for selected values of T and 
T = 650 K,
T = 650 K,
T = 750 K,
 = 500 kg m–3
 = 200 kg m–3
 = 500 kg m–3
p / MPa
0.255 837 018 × 102
0.222 930 643 × 102
0.783 095 639 × 102
h / (kJ kg–1)
0.186 343 019 × 104
0.237 512 401 × 104
0.225 868 845 × 104
u / (kJ kg–1)
0.181 226 279 × 104
0.226 365 868 × 104
0.210 206 932 × 104
s / (kJ kg–1 K–1)
0.405 427 273 × 101
0.485 438 792 × 101
0.446 971 906 × 101
cp / (kJ kg–1 K–1)
0.138 935 717 × 102
0.446 579 342 × 102
0.634 165 359 × 101
w / (m s–1)
0.502 005 554 × 10
3
3
0.383 444 594 × 10
0.760 696 041 × 10
a
3
a It is recommended to verify programmed functions using 8 byte real values for all three combinations of T and
 given in this table.
33
8 Equations for Region 4
This section contains all details relevant for the use of the equations for region 4 of
IAPWS-IF97 (saturation line). Information about the consistency of the basic equation for
this region, the saturation-pressure equation, and the basic equations of regions 1 to 3 at the
corresponding region boundaries is summarized in Section 10. The results of computing-time
comparisons between IAPWS-IF97 and IFC-67 are given in Section 11. The estimates of
uncertainty of the saturation pressure can be found in Section 12.
The equation for describing the saturation line is an implicit quadratic equation which can
be directly solved with regard to both saturation pressure ps and saturation temperature Ts .
This equation reads
 2 2  n1 2  n2  2  n3  2  n4   n5   n6 2  n7  n8  0 ,


(29a)

n9
(29b)
  ps / p
where

and
Ts
T

(29)
1/4
T / T   n
s

10
with p* = 1 MPa and T * = 1 K; for the coefficients n1 to n10 see Table 34.
8.1 The Saturation-Pressure Equation (Basic Equation)
The solution of Eq. (29) with regard to saturation pressure is as follows:

ps 
2C


p
  B  B2  4 AC


4


,
1/2 



where p* = 1 MPa and
A   2  n1  n2
B  n3 2  n4  n5
C  n6 2  n7  n8
with  according to Eq. (29b). The coefficients ni of Eq. (30) are listed in Table 34.
(30)
34
Table 34. Numerical values of the coefficients of the dimensionless saturation
equations, Eqs. (29) to (31)
i
ni
i
1
0.116 705 214 527 67 × 104
6
0.149 151 086 135 30 × 102
2
– 0.724 213 167 032 06 × 106
7
– 0.482 326 573 615 91 × 104
3
– 0.170 738 469 400 92 × 102
8
0.405 113 405 420 57 × 106
4
0.120 208 247 024 70 × 105
9
5
– 0.323 255 503 223 33 × 107
10
ni
– 0.238 555 575 678 49
0.650 175 348 447 98 × 103
Equations (29) to (31) reproduce exactly the p-T values at the triple point according to
Eq. (9), at the normal boiling point [12] and at the critical point according to Eqs. (2) and (3).
Range of validity
Equation (30) is valid along the entire vapor-liquid saturation line from the triple-point
temperature Tt to the critical temperature Tc and can be simply extrapolated to 273.15 K so
that it covers the temperature range
273.15 K  T  647.096 K .
Computer-program verification
To assist the user in computer-program verification of Eq. (30), Table 35 contains test
values for three selected temperatures.
Table 35. Saturation pressures calculated from
Eq. (30) for selected values of T a
T/K
ps / MPa
300
0.353 658 941 × 10–2
500
0.263 889 776 × 101
600
0.123 443 146 × 102
a It is recommended to verify the programmed equation
using 8 byte real values for all three values of T given
in this table.
35
8.2 The Saturation-Temperature Equation (Backward Equation)
The saturation-temperature solution of Eq. (29) reads
1/2
2


Ts n10  D   n10  D   4  n9  n10 D 

2
T
,
(31)
where T * = 1 K and
D
with

2G
 F  F 2  4 EG

1/2
E   2  n3   n6
F  n1 2  n4   n7
G  n2  2  n5   n8
and  according to Eq. (29a). The coefficients ni of Eq. (31) are listed in Table 34.
Range of validity
Equation (31) has the same range of validity as Eq. (30), which means that it covers the
vapor-liquid saturation line according to the pressure range
611.213 Pa  p  22.064 MPa .
The value of 611.213 Pa corresponds to the pressure when Eq. (31) is extrapolated to 273.15 K.
Consistency with the basic equation
Since the saturation-pressure equation, Eq. (30), and the saturation-temperature equation,
Eq. (31), have been derived from the same implicit equation, Eq. (29), for describing the
saturation line, both Eq. (30) and Eq. (31) are completely consistent with each other.
Computer-program verification
To assist the user in computer-program verification of Eq. (31), Table 36 contains test
values for three selected pressures.
36
Table 36. Saturation temperatures calculated from
Eq. (31) for selected values of p a
p / MPa
Ts / K
0.1
0.372 755 919 × 103
1
0.453 035 632 × 103
10
0.584 149 488 × 103
a It is recommended to verify the programmed equation
using 8 byte real values for all three values of p given
in this table.
9 Basic Equation for Region 5
This section contains all details relevant for the use of the equations for region 5 of
IAPWS-IF97. Information about the consistency at the boundary between regions 2 and 5 is
summarized in Section 10. The results of computing-time comparisons between IAPWS-IF97
and IFC-67 are given in Section 11. The estimates of uncertainty of the most relevant
properties can be found in Section 12.
The basic equation for this high-temperature region is a fundamental equation for the
specific Gibbs free energy g. This equation is expressed in dimensionless form,  = g/( RT ),
and is separated into two parts, an ideal-gas part  o and a residual part  r, so that
g  p, T 
RT
   ,    o  ,    r  ,  ,
(32)
where  = p/p* and  = T */T with R given by Eq. (1).
The equation for the ideal-gas part  o of the dimensionless Gibbs free energy reads
6
o
 o  ln    nio Ji ,
(33)
i 1
where  = p/p* and  = T */T with p* = 1 MPa and T * = 1000 K. The coefficients n1o and n2o
were adjusted in such a way that the values for the specific internal energy and specific
entropy in the ideal-gas state relate to Eq. (8). Table 37 contains the coefficients nio and
exponents J io of Eq. (33).
37
Table 37. Numerical values of the coefficients and exponents of the ideal-gas part  o of the
dimensionless Gibbs free energy for region 5, Eq. (33)
i
J io
nio
i
J io
1
0
– 0.131 799 836 742 01 × 102
4
–2
1
2
1
0.685 408 416 344 34 × 10
5
–1
3
–3
– 0.248 051 489 334 66 × 10
6
2
nio
0.369 015 349 803 33
– 0.311 613 182 139 25 × 101
– 0.329 616 265 389 17
The form of the residual part  r of the dimensionless Gibbs free energy is as follows:
6
 r   ni  Ii  Ji ,
(34)
i 1
where  = p/p* and  = T */T with p* = 1 MPa and T * = 1000 K. The coefficients ni and
exponents Ii and Ji of Eq. (34) are listed in Table 38.
All thermodynamic properties can be derived from Eq. (32) by using the appropriate
combinations of the ideal-gas part  o, Eq. (33), and the residual part  r, Eq. (34), of the
dimensionless Gibbs free energy and their derivatives. Relations between the relevant
thermodynamic properties and  o and  r and their derivatives are summarized in Table 39.
All required derivatives of the ideal-gas part and of the residual part of the dimensionless
Gibbs free energy are explicitly given in Table 40 and Table 41, respectively.
Table 38. Numerical values of the coefficients and exponents of the
residual part  r of the dimensionless Gibbs free energy for
region 5, Eq. (34)
i
Ii
Ji
ni
1
1
1
0.157 364 048 552 59 × 10
2
1
2
0.901 537 616 739 44 × 10
3
1
3
 0.502 700 776 776 48 × 10
4
2
3
0.224 400 374 094 85 × 10
5
2
9
 0.411 632 754 534 71 × 10
6
3
7
0.379 194 548 229 55 × 10
38
o
Table 39. Relations of thermodynamic properties to the ideal-gas part  and the residual part  r of
the dimensionless Gibbs free energy and their derivatives a when using Eq. (32)
Property
Relation
Specific volume
v   g/ p T
v( , )
Specific internal energy
u  , 
u  g  T  g/T  p  p(g/ p)T
Specific entropy
RT
s  ,  
s    g/T  p
R
h  ,  
Specific enthalpy
h  g  T  g/T  p
RT
Specific isobaric heat capacity
c p   h /T  p

cv    u /T v
1/2
w  v     p /  v s 

 


   o   r
cv  , 

  2  o   r
 
w2 ( , )

RT
Speed of sound
 
   o   r   o   r
c p  , 
R

   o   r    o   r
R
Specific isochoric heat capacity

p
   o   r
RT
2
r
r 2
(1   r   
)
r
1   2 
1  2 r   2 r 2
1       
1




      
   
r
r
2
2 r
2
a


    
o

o
r
  r 
  2 r 
  r 
  2 r 
  2 r 
r
r
r
r




,
,
,
,





 2


 2 

,




  
  
  
  
   
 r  
  o 
  2 o 
o
 ,     2 
  
  
 o  
39
Table 40. The ideal-gas part  o of the dimensionless Gibbs free
energy and its derivatives a according to Eq. (33)

o
=
6
ln
+
 nio J
o
i
i 1
 o
=
1/
+
0
o
 
=
 1/ 2
+
0
6

=
o
 
=
o
 
=
o
0
+
 nio Jio J
o
i 1
i 1
6
a
0
+
 nio Jio  Jio  1 J
o
i 2
i 1
0
+
0
  o 
  2 o 
  o 
  2 o 
  2 o 
o
o
o
o
 ,     2  ,    
 ,     2  ,    

  
  
  
  
   
 o  
Table 41. The residual part  r of the dimensionless Gibbs free energy and its
derivatives a according to Eq. (34)
6
 r   ni  Ii  Ji
i 1
6
6
 r   ni Ii  Ii 1  Ji
r
 
  ni I i  I i  1  Ii 2  Ji
i 1
i 1
6
 r   ni  Ii Ji  Ji 1
 r 
i 1
6
 ni  I Ji  Ji  1  J 2
i
i
i 1
6
r
 
  ni I i  Ii 1Ji  Ji 1
i 1
a
  r 
  2 r 
  r 
  2 r 
  2 r 
r
r
r
r








,
,
,
,

 2


 2 






  
  
  
  
   
 r  
40
Range of validity
Equation (32) covers region 5 of IAPWS-IF97 defined by the following temperature and
pressure range:
1073.15 K  T  2273.15 K
0  p  50 MPa .
Equation (32) is only valid for pure undissociated water; any dissociation will have to be
taken into account separately.
Computer-program verification
To assist the user in computer-program verification of Eq. (32), Table 42 contains test
values of the most relevant properties.
Table 42. Thermodynamic property values calculated from Eq. (32) for selected values of T and p a
T = 1500 K,
p = 0.5 MPa
T = 1500 K,
p = 30 MPa
T = 2000 K,
p = 30 MPa
v / (m3 kg–1)
0.138 455 090 × 101
0.230 761 299 × 10–1
0.311 385 219 × 10–1
h / (kJ kg–1)
0.521 976 855 × 104
0.516 723 514 × 104
0.657 122 604 × 104
u / (kJ kg–1)
0.452 749 310 × 104
0.447 495 124 × 104
0.563 707 038 × 104
s / (kJ kg–1 K–1)
0.965 408 875 × 101
0.772 970 133 × 101
0.853 640 523 × 101
cp / (kJ kg–1 K–1)
0.261 609 445 × 101
0.272 724 317 × 101
0.288 569 882 × 101
w / (m s–1)
0.917 068 690 × 103
0.928 548 002 × 103
0.106 736 948 × 104
a It is recommended to verify programmed functions using 8 byte real values for all three combinations of T and
p given in this table.
10 Consistency at Region Boundaries
For any calculation of thermodynamic properties of water and steam across the region
boundaries, the equations of IAPWS-IF97 have to be sufficiently consistent along the
corresponding boundary. For the properties considered in this respect, this section presents the
achieved consistencies in comparison to the permitted inconsistencies according to the socalled Prague values [13].
10.1 Consistency at Boundaries between Single-Phase Regions
For the boundaries between single-phase regions, the consistency investigations were
performed for the following basic equations and region boundaries; see Fig. 1:
 Equations (7) and (28) along the 623.15 K isotherm for pressures from 16.53 MPa ( ps from
Eq. (30) for 623.15 K) to 100 MPa corresponding to the boundary between regions 1 and 3.
41
 Equations (15) and (28) with respect to the boundary between regions 2 and 3 defined by
the B23-equation, Eq. (5), for temperatures between 623.15 K and 863.15 K.
 Equations (15) and (32) with respect to the 1073.15 K isotherm for p  50 MPa corresponding to the boundary between regions 2 and 5.
The results of the consistency investigations for these three region boundaries are
summarized in Table 43. In addition to the permitted inconsistencies corresponding to the
Prague values [13], the actual inconsistencies characterized by their maximum and root-meansquare values,  xmax and  x RMS , along the three boundaries are given for x = v, h, cp , s, g,
and w. It can be seen that the inconsistencies between the basic equations along the
corresponding region boundaries are extremely small.
Table 43. Inconsistencies between basic equations for single-phase regions at the joint region
boundary
Inconsistency
x
v/%
1
 h/(kJ kg )
Prague
value
1
 s/(J kg K )
1
g/(kJ kg )
Regions 2/3
Eqs. (15)/(28)
Regions 2/5
Eqs. (15)/(32)
 xmax  x RMS a  xmax  x RMS a  xmax  x RMS a
0.05
0.004
0.002
0.018
0.007
0.012
0.007
0.2
0.031
0.014
0.134
0.073
0.096
0.070
1
0.195
0.058
0.353
0.169
0.074
0.049
0.2
0.042
0.022
0.177
0.094
0.142
0.084
0.2
0.005
0.005
0.005
0.003
0.087
0.072
1b
0.299
0.087
0.403
0.073
0.046
0.028
cp/%
1
Regions 1/3
Eqs. (7)/(28)
w/%
a The  x
RMS values (see Nomenclature) were calculated from about 10 000 points evenly distributed along the
corresponding boundary.
b The permitted inconsistency value for w is not included in the Prague values [13].
10.2 Consistency at the Saturation Line
The consistency investigations along the vapor-liquid saturation line were performed for
the properties ps , Ts , and g. The calculations concern the following basic equations and
ranges of the saturation line, see Fig. 1; the way of calculating the inconsistencies  ps , Ts ,
and g is also given:
 Equations (7), (15) and (30) on the saturation line for temperatures from Tt = 273.15 K to
623.15 K.
42
 ps = ps, Eq.(7), Eq.(15)  ps, Eq.(30)
Ts = Ts, Eq.(7), Eq.(15)  Ts, Eq.(31)
g = g Eq.(7)
g Eq.(15)
(35a)
(35b)
(35c)
The calculation of ps and of Ts from Eqs. (7) and (15) is made via the Maxwell criterion
(phase-equilibrium condition) for given values of T or p. The g values are determined for
given T values and corresponding ps values from Eq. (30).
 Equations (28) and (30) on the saturation line for temperatures from 623.15 K to
Tc = 647.096 K.
 ps = ps, Eq.(28)
 ps, Eq.(30)
Ts = Ts, Eq.(28)
 Ts, Eq.(31)
g = gEq.(28), Eq.(30)  gEq.(28), Eq.(30)
(36a)
(36b)
(36c)
The calculation of ps and Ts from Eq. (28) is made via the Maxwell criterion for given
temperatures or pressures, respectively. The inconsistency g corresponds to the difference
g(  , T )  g(  , T ) which is calculated from Eq. (28) after   and   are determined
from Eq. (28) by iteration for given T values and corresponding ps values from Eq. (30).
 Equations (7), (15) and (28) on the saturation line at 623.15 K. This is the only point on the
saturation line where the validity ranges of the fundamental equations of regions 1 to 3
meet each other.
 ps = ps, Eq.(7), Eq.(15)  ps, Eq.(28)
(37a)
Ts = Ts, Eq.(7), Eq.(15)  Ts, Eq.(28)
(37b)
g = g Eq.(7), Eq.(15)
 g Eq.(28)
(37c)
All three properties ps and Ts and g are calculated via the Maxwell criterion from the
corresponding equations.
The results of these consistency investigations along the saturation line are summarized in
Table 44. In addition to the permitted inconsistencies corresponding to the Prague values
[13], the actual inconsistencies characterized by their maximum and root-mean-square values,
 xmax and  x RMS , for the two sections of the saturation line are given for x = ps , Ts and g.
It can be seen that the inconsistencies between the basic equations for the corresponding
single-phase region and the saturation-pressure equation are extremely small. This statement
also holds for the fundamental equations, Eqs. (7), (15), and (28), among one another and not
only in relation to the saturation-pressure equation, Eq. (30), see the last column in Table 44.
43
Table 44. Inconsistencies between the basic equations valid at the saturation line
Inconsistency
x
Prague
value
Tt  T  623.15 K
Eqs. (7),(15)/(30)
623.15 K  T  Tc
Eqs. (28)/(30)
 xmax  xRMS a
 xmax  xRMS a
T = 623.15 K
Eqs. (7),(15)/(28)
 ps/%
0.05
0.0069
0.0033
0.0026
0.0015
0.0041
 Ts/%
0.02
0.0006
0.0003
0.0003
0.0002
0.0006
0.2
0.012
0.006
0.002
0.001
0.005
g/(kJ kg1)
a The  x
RMS values (see Nomenclature) were calculated from about 3000 points evenly distributed along the
two sections of the saturation line.
11 Computing Time of IAPWS-IF97 in Relation to IFC-67
A very important requirement for IAPWS-IF97 was that its computing speed in relation to
IFC-67 should be significantly faster. The computation-speed investigations of IAPWS-IF97
in comparison with IFC-67 are based on a special procedure agreed to by IAPWS.
The computing times were measured with a benchmark program developed by IAPWS;
this program calculates the corresponding functions at a large number of state points well
distributed proportionately over each region. The test configuration agreed on was a
PC Intel 486 DX 33 processor and the MS Fortran 5.1 compiler. The relevant functions of
IAPWS-IF97 were programmed with regard to short computing times. The calculations with
IFC-67 were carried out with the ASME program package [14] speeded up by excluding all
parts which were not needed for these special benchmark tests.
The measured computing times were used to calculate computing-time ratios
IFC-67 / IAPWS- IF97, called CTR values in the following. These CTR values, determined in
a different way for regions 1, 2, and 4 (see Section 11.1) and for regions 3 and 5 (see
Section 11.2), are the characteristic quantities for the judgment of how much faster the
calculations with IAPWS-IF97 are in comparison with IFC-67. Metastable states are not
included in these investigations.
11.1 Computing-Time Investigations for Regions 1, 2, and 4
The computing-time investigations for regions 1, 2, and 4, which are particularly relevant
to computing time, were performed for the functions listed in Table 45. Each function is
associated with a frequency-of-use value. Both the selection of the functions and the values
for the corresponding frequency of use are based on a worldwide survey made among the
power plant companies and related industries.
For the computing-time comparison between IAPWS-IF97 and IFC-67 for regions 1, 2,
and 4, the total CTR value of these three regions together was the decisive criterion, where
44
the frequencies of use have to be taken into account. The total CTR value was calculated as
follows: As has been described before, the computing times for each function were
determined for IFC-67 and for IAPWS-IF97. Then, these values were weighted by the
corresponding frequencies of use and added up for the 16 functions of the three regions. The
total CTR value is obtained from the sum of the weighted computing times for IFC-67 divided
by the corresponding value for IAPWS-IF97. The total CTR value for regions 1, 2, and 4
amounts to
(38)
CTR regions 1, 2, 4 = 5.1 .
This means that for regions 1, 2, and 4 together the property calculations with IAPWS-IF97
are more than five times faster than with IFC-67.
Table 45. Results of the computing-time investigations of IAPWSIF97 in relation to IFC-67 for regions 1, 2, and 4 a
Region
1
b
Function
Frequency
of use
%
Computing-time
ratio
IFC-67 / IF97
v ( p, T )
h ( p, T )
T ( p, h )
h ( p, s )
2.9
9.7
3.5
1.2
2.7
2.9
24.8
10.0
 region 1: 5.6 c

2
v ( p, T )
h ( p, T )
s ( p, T )
T ( p, h )
v ( p, h )
s ( p, h )
T ( p, s )
h ( p, s )
6.1
12.1
1.4
8.5
3.1
1.7
1.7
4.9
2.1
2.9
1.4
12.4
6.4
4.2
8.1
5.6
 region 2: 5.0 c

4
ps( T )
Ts( p )
h ( p )
h ( p )
8.0
30.7
2.25
2.25
1.7
5.6
4.4
4.2

 region 4: 4.9 c

 regions 1, 2 and 4: 5.1 c
a Based on the agreed PC Intel 486 DX 33 with MS Fortran 5.1 compiler.
b For the definition of the regions see Fig. 1.
c This CTR value is based on the computing times for the single
functions weighted by the frequency-of-use values; see text.
45
Table 45 also contains total CTR values separately for each of regions 1, 2, and 4. In
addition, CTR values for each single function are given. When using IAPWS-IF97 the
functions depending on p,h and p,s for regions 1 and 2 and on p for region 4 were calculated
from the backward equations alone (functions explicit in T ) or from the basic equations in
combination with the corresponding backward equation.
If a faster processor than specified above is used for the described benchmark tests, similar
CTR values are obtained. A corresponding statement is also valid for other compilers than the
specified one.
11.2 Computing-Time Investigations for Region 3
For regions 3 and 5 the CTR values only relate to single functions and are given by the
quotient of the computing time needed for IFC-67 calculation and the computing time when
using IAPWS-IF97; there are no frequency-of-use values for functions relevant to these two
regions.
For region 3 of IAPWS-IF97, corresponding to regions 3 and 4 of IFC-67, the computingtime investigations relate to the functions p ( v, T ), h ( v, T ), cp ( v, T ), and s ( v, T ) where 10 %
of the test points are in region 4 of IFC-67.
Table 46 lists the CTR values obtained for the relevant functions of region 3. Roughly
speaking, IAPWS-IF97 is more than three times faster than IFC-67 for this region.
Table 46. Results of the computing-time investigations of IAPWS-IF97 in
relation to IFC-67 for region 3a
Function
p ( v, T )
h ( v, T )
cp ( v , T )
s ( v, T )
Computing time
ratio
IFC-67 / IF97
3.8
4.3
2.9
3.2
a Based on the agreed PC Intel 486 DX 33 with
MS Fortran 5.1 compiler.
46
12 Estimates of Uncertainties
Estimates have been made of the uncertainty of the specific volume, specific isobaric heat
capacity, speed of sound, and saturation pressure when calculated from the corresponding
equations of IAPWS-IF97. These estimates were derived from the uncertainties of
IAPWS-95 [3], from which the input values for fitting the IAPWS-IF97 equations were
calculated, and in addition by taking into account the deviations between the corresponding
values calculated from IAPWS-IF97 and IAPWS-95. Since there is no reasonable basis for
estimating the uncertainty of specific enthalpy (because specific enthalpy is dependent on the
selection of the zero point, only enthalpy differences of different size are of interest), no
uncertainty is given for this property. However, the uncertainty of isobaric enthalpy differences
is smaller than the uncertainty in the isobaric heat capacity.
For the single-phase region, tolerances are indicated in Figs. 3 to 5 which give the estimated
uncertainties in various areas. As used here "tolerance" means the range of possible values as
judged by IAPWS, and no statistical significance can be attached to it. With regard to the
uncertainty for the speed of sound and the specific isobaric heat capacity, see Figs. 4 and 5, it
should be noted that the uncertainties for these properties increase drastically when approaching
the critical point. The statement "no definitive uncertainty estimates possible" for temperatures
above 1273 K is based on the fact that this range is beyond the range of validity of IAPWS-95
and the corresponding input values for IAPWS-IF97 were extrapolated from IAPWS-95. From
various tests of IAPWS-95 [3] it is expected that these extrapolations yield reasonable values.
For the saturation pressure, the estimate of uncertainty is shown in Fig. 6.
Estimated uncertainties in enthalpy, in enthalpy differences in the single-phase region, and
in the enthalpy of vaporization are given in IAPWS Advisory Note No. 1 [15].
47
Fig. 3. Uncertainties in specific volume, v v, estimated for the corresponding equations of IAPWS-IF97. In
the enlarged critical region (triangle), the uncertainty is given as percentage uncertainty in pressure,
p/p. This region is bordered by the two isochores 0.0019 m3 kg and 0.0069 m3 kg and by the
30 MPa isobar. The positions of the lines separating the uncertainty regions are approximate.
Fig. 4. Uncertainties in specific isobaric heat capacity, cp cp, estimated for the corresponding equations of
IAPWS-IF97. For the definition of the triangle around the critical point, see Fig. 3. The positions of the
lines separating the uncertainty regions are approximate.
48
Fig. 5. Uncertainties in speed of sound, ww, estimated for the corresponding equations of IAPWS-IF97. For
the definition of the triangle around the critical point, see Fig. 3. The positions of the lines separating the
uncertainty regions are approximate.
Fig. 6. Uncertainties in saturation pressure, ps ps, estimated for the saturation-pressure equation, Eq. (30).
49
13 References
[1]
International Formulation Committee of the 6th International Conference on the Properties of
Steam, The 1967 IFC Formulation for Industrial Use, Verein Deutscher Ingenieure, Düsseldorf,
1967.
[2]
Wagner, W., Cooper, J. R., Dittmann, A., Kijima, J., Kretzschmar, H.-J., Kruse, A., Mareš, R.,
Oguchi, K., Sato, H., Stöcker, I., Šifner, O., Takaishi, Y., Tanishita, I., Trübenbach, J., and
Willkommen, Th., The IAPWS Industrial Formulation 1997 for the Thermodynamic Properties
of Water and Steam, J. Eng. Gas Turbines & Power 122, 150-182 (2000).
[2a] Wagner, W., Dauber, F., Kretzschmar, H.-J., Mareš, R., Miyagawa, K., Parry, W. T., and
Span, R., The New Basic Equation for the Extended Region 5 of the Industrial Formulation
IAPWS-IF97 for Water and Steam. To be submitted to J. Eng. Gas Turbines & Power.
[2b] IAPWS Advisory Note No. 2: Roles of Various IAPWS Documents Concerning the Thermodynamic Properties of Ordinary Water Substance (2009). Available from http://www.iapws.org
[3]
IAPWS, Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of
Ordinary Water Substance for General and Scientific Use (2009). Available from
http://www.iapws.org
[4]
Cohen, E. R. and Taylor, B. N., The 1986 Adjustment of the Fundamental Physical Constants,
CODATA Bulletin No. 63, Committee on Data for Science and Technology, Int. Council of
Scientific Unions, Pergamon Press, Oxford, 1986.
[5]
Audi, G. and Wapstra, A. H., The 1993 atomic mass evaluation, (I) Atomic mass table, Nuclear
Physics A 565, 1-65 (1993).
[6]
IUPAC Commission on the Atomic Weights and Isotopic Abundances, Subcommittee for
Isotopic Abundance Measurements, Isotopic compositions of the elements 1989, Pure Appl.
Chem. 63, 991-1002 (1991).
[7]
IAPWS, Release on the Values of Temperature, Pressure and Density of Ordinary and Heavy
Water Substances at Their Respective Critical Points (1992). Available from
http://www.iapws.org
[8]
Preston-Thomas, H., The International Temperature Scale of 1990 (ITS-90), Metrologia 27, 310 (1990).
[9]
Guildner, L. A., Johnson, D. P., and Jones, F. E., Vapor Pressure of Water at Its Triple Point,
J. Res. Natl. Bur. Stand. 80A, 505-521 (1976).
[10] IAPWS, Revised Release on the Pressure along the Melting and Sublimation Curves of
Ordinary Water Substance (2008). Available from http://www.iapws.org
[11] Wagner, W. and Pruß, A., The IAPWS Formulation 1995 for the Thermodynamic Properties of
Ordinary Water Substance for General and Scientific Use, J. Phys. Chem. Ref. Data 31, 387535 (2002).
[12] IAPWS, Revised Supplementary Release on Saturation Properties of Ordinary Water Substance
(1992). Available from http://www.iapws.org
[13] Minutes of the meetings of the International Formulation Committee of ICPS in Prague, 1965.
[14] McClintock, R. B. and Silvestri, G. J., Formulations and iterative procedures for the
calculation of properties of steam, The American Society of Mechanical Engineers, New York,
1968.
[15] IAPWS Advisory Note No. 1: Uncertainties in Enthalpy for the IAPWS Formulation 1995 for
the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use
(IAPWS-95) and the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of
Water and Steam (IAPWS-IF97) (2003). Available from http://www.iapws.org